Folds

The accumulate procedure is also known as fold-right , because it combines the first element of the sequence with the result of combining all the elements to the right. There is also a fold-left , which is similar to fold-right , except that it combines elements working in the opposite direction:

(define (fold-left op initial sequence)
  (define (iter result rest)
    (if (null? rest)
        result
        (iter (op result (car rest))
              (cdr rest))))
  (iter initial sequence))

What are the values of

(fold-right / 1 (list 1 2 3))

(fold-left / 1 (list 1 2 3))

(fold-right list nil (list 1 2 3))

(fold-left list nil (list 1 2 3))

Give a property that op should satisfy to guarantee that fold-right and fold-left will produce the same values for any sequence.


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(define example (list 1 2 3))

(check-equal? (fold-right / 1 example) 3/2)
(check-equal? (fold-left / 1 example) 1/6)
(check-equal? (fold-right list '() example) '(1 (2 (3 ()))))
(check-equal? (fold-left list '() example) '(((() 1) 2) 3))
(check-equal? (fold-right + 0 example) (fold-left + 0 example))
(check-equal? (fold-right * 1 example) (fold-left * 1 example))